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A002162
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Decimal expansion of natural logarithm of 2.
(Formerly M4074 N1689)
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40
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6, 9, 3, 1, 4, 7, 1, 8, 0, 5, 5, 9, 9, 4, 5, 3, 0, 9, 4, 1, 7, 2, 3, 2, 1, 2, 1, 4, 5, 8, 1, 7, 6, 5, 6, 8, 0, 7, 5, 5, 0, 0, 1, 3, 4, 3, 6, 0, 2, 5, 5, 2, 5, 4, 1, 2, 0, 6, 8, 0, 0, 0, 9, 4, 9, 3, 3, 9, 3, 6, 2, 1, 9, 6, 9, 6, 9, 4, 7, 1, 5, 6, 0, 5, 8, 6, 3, 3, 2, 6, 9, 9, 6, 4, 1, 8, 6, 8, 7
(list; constant; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| Newton calculated the first 16 terms of this sequence.
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REFERENCES
| S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3.3.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. W. Sweeney, On the computation of Euler's constant, Math. Comp., 17 (1963), 170-178.
Uhler, Horace S.; Recalculation and extension of the modulus and of the logarithms of 2, 3, 5, 7 and 17. Proc. Nat. Acad. Sci. U. S. A. 26, (1940). 205-212.
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LINKS
| Harry J. Smith, Table of n, a(n) for n = 0..20000
D. H. Bailey and J. M. Borwein, Experimental Mathematics: Examples, Methods and Implications
Paul Cooijmans, Odds.
X. Gourdon and P. Sebah, The logarithm constant:log(2)
I. Newton, The method of fluxions and infinite series; with its application to the geometry of curve-lines, 1736; see p. 96.
S. Plouffe, log(2), natural logarithm of 2 to 2000 places
S. Ramanujan, Question 260, J. Ind. Math. Soc.
Eric Weisstein's World of Mathematics, Natural Logarithm of 2, Masser-Gramain Constant, Logarithmic Integral
Wikipedia, Natural logarithm of 2. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 02 2010]
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FORMULA
| log(2) = Sum_{ k >= 1 } 1/(k*2^k) = Sum_{j >= 1} (-1)^(j+1)/j.
log(2) = Integral_{t = 0..1 } dt/(1+t).
log(2) = 2/3 * (1 + Sum{k=1..inf, 2/[(4k)^3-4k]}) (Ramanujan).
log(2)=4*sum_{k=0..inf} [3-2*sqrt(2)]^(2k+1)/(2k+1) (Y. Luke) - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 13 2006
log(2) = 1-(1/2)Sum_{ k >= 1 } 1/(k*(2k+1)) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Jan 06 2009, Jan 08 2009]
log(2) = 4*Sum_{ k >= 0 } 1/((4k+1)(4k+2)(4k+3)) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Jan 08 2009]
log(2) = 7/12+24*sum_{k=1..infinity} 1/(A052787(k+4)*A000079(k)). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 23 2009]
From Alexander R. Povolotsky, Jul 04 2009: (Start)
log(2) = 1/4*(3 - sum(1/(n*(n+1)*(2*n+1)), n=1...infinity))
log(2)=(230166911/9240-Sum((1/2)^k* (11/k+10/(k+1)+9/(k+2)+8/(k+3)+7/(k+4)+6/(k+5)-6/(k+7)-7/(k+8)-8/(k+9) -9/(k+10)-10/(k+11)), k = 1 .. infinity))/35917 (End)
log(2) = A052882/A000670. [From Mats Granvik (mats.granvik(AT)abo.fi), Aug 10 2009]
From log(1-x-x^2) at x=1/2, log(2)=(1/2)*Sum_{k>=1}L(k)/(k*2^k), where L(n) is the n-th Lucas number (A000032). [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Oct 24 2009]
log(2) = Sum_{ k >= 1 } 1/(cos(k*Pi/3)*k*2^k) (Cf. A176900) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Apr 29 2010]
log(2) =(sum(1/(n^2*(n+1)^2*(2*n+1)),n=1...infinity) +11)/16 [From Alexander R. Povolotsky (pevnev(AT)juno.com),Jan 13, 2011]
log(2) =sum((2*n+1)/(sum(k^2,k=1...n)^2),n=1...infinity) +396)/576 [From Alexander R. Povolotsky (pevnev(AT)juno.com),Jan 14, 2011]
log(2) = 1/4*(3 - sum(1/(n*(n+1)*(2*n+1)),n=1...infinity)) [From Alexander R. Povolotsky (pevnev(AT)juno.com), Dec 16 2008]
log(2) = 1/4*(3 - sum(1/(n*(n+1)*(2*n+1)),n=1...infinity)) - from and by Alexander R. Povolotsky [From Alexander R. Povolotsky (pevnev(AT)juno.com), Dec 16 2008]
log(2) = 105*(sum(1/(2*n*(2*n+1)*(2*n+3)*(2*n+5)*(2*n+7)),n=1...infinity) - 319/44100) log(2) = (319/420 - 3/2*sum(1/(6*n^2+39*n+63),n=1...infinity)) [From Alexander R. Povolotsky (pevnev(AT)juno.com), Dec 16 2008]
log(2) = sum(k>=1, A191907(2,k)/k ). [From Mats Granvik (mats.granvik(AT)abo.fi),Jun 19, 2011]
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EXAMPLE
| 0.693147180559945309417232121458176568075500134360255254120680009493393...
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MATHEMATICA
| RealDigits[N[Log[2], 200]][[1]] (*From Vladimir Joseph Stephan Orlovsky, Feb 21 2011*)
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PROG
| (PARI) { default(realprecision, 20080); x=10*log(2); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b002162.txt", n, " ", d)); } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Apr 21 2009]
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CROSSREFS
| Cf. A016730 Continued fraction.
Sequence in context: A129938 A022698 A013707 * A072365 A085138 A153872
Adjacent sequences: A002159 A002160 A002161 * A002163 A002164 A002165
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KEYWORD
| cons,nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Fixed my PARI program, had -n Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 19 2009
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